Gcd-closed sets and divisibility among power LCM matrices

Abstract

Let $a,b$ and $n$ be positive integers and let $S=\{x_1, \cdots, x_n\}$ be a set of $n$ distinct positive integers. For $x\in S$, one defines $G_{S}(x)=\{d\in S: d<x, d|x \ {\rm and} \ (d|y|x, y\in S)\Rightarrow y\in \{d,x\}\}$. We denote by $(S^a)$ (resp. $[S^a]$) the $n\times n$ matrix having the $a$th power of the greatest common divisor (resp. the least common multiple) of $x_i$ and $x_j$ as its $(i,j)$-entry. In 1995, Bourque and Ligh showed that the $a$th power GCD matrix $(S^a)$ divides the $a$th power LCM matrix $[S^a]$ in the ring $M_n(\mathbb Z)$ of $n\times n$ matrices over the integers when $S$ is FC. In 2002, Hong proved that such factorization is no longer true when $S$ is gcd closed. In 2008 (resp. 2026), Hong showed that $[S^a]\mid [S^b]$ if $a\mid b$ and $S$ is a divisor chain (resp. an FC set). In this paper, we show that for arbitrary positive integers $a$ and $b$ with $a|b$, the $b$th power matrices $[S^b]$ is divisible by the $a$th power matrix $[S^a]$ if $S$ is a gcd-closed set (i.e. $\gcd(x_i, x_j)\in S$ for all integers $i$ and $j$ with $1\le i, j\le n$) such that the condition $\mathcal G$ is satisfied (i.e., for any $x\in S$, either $G_S(x)$ contains at most one elements, or $G_S(x)$ contains at least two elements and satisfies that $[y_1,y_2]=x$ and $(y_1,y_2)\in G_S(y_1)\cap G_S(y_2)$) for any $\{y_1,y_2\}\subseteq G_S(x)$. This confirms a conjecture of Hong proposed in [S.F. Hong, Divisibility among power GCD matrices and power LCM matrices, {\it Bull. Aust. Math. Soc.} {\bf 113} (2026), 231-243].